×

The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space. (English) Zbl 0928.47041

Recently, C.-H. Chu and P. Mellon [Proc. Am. Math. Soc. 125, No. 6, 1771-1777 (1997; Zbl 0871.46021)] proved that the well-known Denjoy-Wolff theorem is valid for a compact holomorphic fixed-point-free self-map of the open unit ball of a Hilbert space. In the present paper, the authors prove that such a result is valid in a strictly convex Banach space and for \(k_B\)-nonexpansive mappings on the unit ball of uniformly convex Banach space, namely: Let \(X\) be a complex strictly convex (or: uniformly convex) Banach space with an open unit ball \(B\). For each compact, holomorphic (or: compact, \(k_B\)-nonexpansive) and fixed-point-free mapping \(f: B\to B\) there exists \(\xi\in\partial B\) such that the sequence \(\{f^n\}\) of iterates of \(f\) converges locally uniformly on \(B\) to the constant map taking the value \(\xi\).

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B20 Geometry and structure of normed linear spaces

Citations:

Zbl 0871.46021
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abate, M., Horospheres and iterates of holomorphic maps, Math. Z., 198, 225-238 (1988) · Zbl 0628.32035
[2] Abate, M., Iteration theory, compactly divergent sequences and commuting holomorphic maps, Ann. Scuola Norm. Sup. Pisa, 18, 167-191 (1991) · Zbl 0760.32014
[3] Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifolds (1989), Mediterranean Press · Zbl 0747.32002
[4] Burckel, R. B., Iterating analytic self-maps of discs, Am. Math. Mon., 88, 396-407 (1981) · Zbl 0466.30001
[5] Calka, A., On conditions under which isometries have bounded orbits, Colloq. Math., 48, 219-227 (1984) · Zbl 0558.54021
[6] Chae, S. B., Holomorphy and Calculus in Normed Spaces (1985), Dekker: Dekker New York · Zbl 0571.46031
[7] Chu, C.-H.; Mellon, P., Iteration of compact holomorphic maps on a Hilbert ball, Proc. Amer. Math. Soc., 125, 1771-1777 (1997) · Zbl 0871.46021
[8] Denjoy, A., Sur l’iteration des fonctions analytiques, C.R. Acad. Sc. Paris, 182, 255-257 (1926) · JFM 52.0309.04
[9] Dineen, S.; Timoney, R. M.; Vigué, J.-P., Pseudodistances invariantes sur les domaines d’un espace localement convexe, Ann. Scuola Norm. Sup. Pisa, 12, 515-529 (1985) · Zbl 0603.46052
[10] Earle, C. J.; Hamilton, R. S., A fixed point theorem for holomorphic mappings, Proc. Symp. Pure Math. (1970), Amer. Math. Soc: Amer. Math. Soc Providence, p. 61-65 · Zbl 0205.14702
[11] Edelstein, M., The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc., 78, 206-208 (1972) · Zbl 0231.47029
[12] Franzoni, T.; Vesentini, E., Holomorphic Maps and Invariant Distances (1980), North-Holland: North-Holland Amsterdam · Zbl 0447.46040
[13] Goebel, K., Fixed points and invariant domains of holomorphic mappings of the Hilbert ball, Nonlinear Anal., 6, 1327-1334 (1982) · Zbl 0525.47039
[14] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0708.47031
[15] Goebel, K.; Reich, S., Iterating holomorphic self-mappings of the Hilbert ball, Proc. Japan Acad., 58, 349-352 (1982) · Zbl 0543.47049
[16] Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings (1984), Dekker: Dekker New York · Zbl 0537.46001
[17] Goebel, K.; Sekowski, T.; Stachura, A., Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal., 4, 1011-1021 (1980) · Zbl 0448.47048
[18] Harris, L. A., Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy (1979), North Holland: North Holland Amsterdam, p. 345-406
[19] Hervé, M., Quelques propriétés des applications analytiques d’une boule à m dimensions dans elle-même, J. Math. Pures et Appl., 42, 117-147 (1963) · Zbl 0116.28903
[20] Kryczka, A.; Kuczumow, T., The Denjoy-Wolff-type theorem for compact\(k_{B_H} \), Ann. Univ. Mariae Curie-Sklodowska Sect. A, 51, 179-183 (1997) · Zbl 1012.47038
[21] Kubota, Y., Iteration of holomorphic maps of the unit ball into itself, Proc. Amer. Math. Soc., 88, 476-480 (1983) · Zbl 0518.32016
[22] Kuczumow, T.; Stachura, A., Iterates of holomorphic and\(k_D\)-nonexpansive mappings in convex domains in \(C^n \), Adv. Math., 81, 90-98 (1990) · Zbl 0726.32016
[23] Lempert, L., Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math., 8, 257-261 (1982) · Zbl 0509.32015
[24] MacCluer, B. D., Iterates of holomorphic self-maps of the open unit ball in \(C^n\), Michigan Math. J., 30, 97-106 (1983) · Zbl 0528.32019
[25] Reich, S., Averaged mappings in the Hilbert ball, J. Math. Anal. Appl., 109, 199-206 (1985) · Zbl 0588.47061
[26] Rudin, W., Function Theory on the Unit Ball in \(C^n (1980)\), Springer-Verlag: Springer-Verlag Berlin
[27] Schauder, J., Der Fixpunktsatz in Funktionalräumen, Studia Math., 2, 171-180 (1930) · JFM 56.0355.01
[28] Stachura, A., Iterates of holomorphic self-maps of the unit ball in Hilbert spaces, Proc. Amer. Math. Soc., 93, 88-90 (1985) · Zbl 0607.47058
[29] Thorp, E.; Whitley, R., The strong maximum modulus theorem for analytic functions into a Banach space, Proc. Amer. Math. Soc., 18, 640-646 (1967) · Zbl 0185.20102
[30] Vesentini, E., Invariant distances and invariant differential metrics in locally convex spaces, Spectral Theory, 8, 493-512 (1982)
[31] Vesentini, E., Su un teorema di Wolff e Denjoy, Rend. Sem. Mat. Fis. Milano, 53, 17-25 (1983) · Zbl 0596.30038
[32] Vesentini, E., Iteration of holomorphic maps, Russ. Math. Surveys, 40, 7-11 (1985) · Zbl 0596.30037
[33] Wolff, J., Sur l’iteration des fonctions bornees, C.R. Acad. Sc. Paris, 182, 42-43 (1926) · JFM 52.0309.02
[34] P. Yang, Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains, 1978; P. Yang, Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains, 1978
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.